Example 3.5.6.9 (054C)

Example 3.5.6.9. Let $n \geq 0$ be an integer and let $X = \mathrm{K}( A_{\ast } )$ be the Eilenberg-MacLane space associated to a chain complex of abelian groups

\[ \cdots \rightarrow A_{n+1} \xrightarrow {\partial } A_{n} \xrightarrow {\partial } A_{n-1} \xrightarrow { \partial } A_{n-2} \rightarrow \cdots \]

(see Construction 2.5.6.3). Let $B_{n} \subseteq A_{n}$ denote the image of the differential $\partial : A_{n+1} \rightarrow A_{n}$, and let $A'_{\ast }$ denote the chain complex

\[ \cdots \rightarrow 0 \rightarrow A_{n} / B_{n} \rightarrow A_{n-1} \xrightarrow { \partial } A_{n-2} \rightarrow \cdots \]

Since $A'_{\ast }$ is concentrated in degrees $\leq n$, the Eilenberg-MacLane space $\mathrm{K}( A'_{\ast } )$ is $n$-groupoid, which we can identify with the fundamental $n$-groupoid $\pi _{\leq n}(X)$. More precisely, the quotient map $A_{\ast } \twoheadrightarrow A'_{\ast }$ is an isomorphism in degrees $< n$ and induces an isomorphism on homology in degrees $\leq n$, so the induced map of Kan complexes $\mathrm{K}( A_{\ast } ) \twoheadrightarrow \mathrm{K}( A'_{\ast } )$ exhibits $\mathrm{K}(A'_{\ast })$ as a fundamental $n$-groupoid of $X$.